求函数3x/x²+x+1的取值范围

y=3x/(x^2+x+1)=3/(x+1+1/x) 当x≠0时
∵x^+x+1=(x+1/2)^2+3/4>0
∴函数定义域为R

x>0时
x+1/x>=2
x+1+1/x>=3
0<1/(x+1+1/x)<=1/3
0<3/(x+1+1/x)<=1

x<0时
-x>0 -1/x>0
(-x)+(-1/x)>=2
-x-1-1/x>=1
0<1/(-x-1-1/x)<=1
-1<=1/(x+1+1/x)<0
-3<=3/(x+1+1/x)<0

x=0时
3x/(x^2+x+1)=0

∴函数的取值范围是[-3,1]